There are less than three weeks before the government could careen off this ‘‘fiscal cliff,’’ but the chief GOP negotiator, House Speaker John Boehner, Republican of Ohio, said Wednesday that ‘‘serious differences’’ remain between him and President Obama after an exchange of offers and a pair of conversations this week.

Boehner spoke after a closed-door meeting with fellow GOP lawmakers in which he advised them not to make plans for the week after Christmas.

Neither side has given much ground, and his exchange of proposals with Obama seemed to generate hard feelings more than progress. The White House has slightly reduced its demands on taxes — from $1.6 trillion over a decade to $1.4 trillion — but isn’t yielding on demands that rates rise for wealthier earners.

Boehner responded with an offer very much like one he gave the White House more than a week ago that offered $800 billion in new revenue, half of Obama’s demand. Boehner is also pressing for an increase in the Medicare eligibility age and a stingier cost-of-living adjustment for Social Security recipients.

Boehner spokesman Michael Steel said the two men did not have any follow-up talks Wednesday.

‘‘There were some offers that were exchanged back and forth yesterday, and the president and I had a pretty frank conversation about just how far apart we are,’’ Boehner said after his meeting with fellow Republican lawmakers.

‘‘He said it’s looking like trench warfare,’’ said Representative John J. Duncan Jr., Republican of Tennessee, referring to Boehner.